I don't speak Italian so could not watch the video with confidence

as to what the message is. I don't know what he is saying

nor can read the out of focus chalk marks.

I translate the words of the professor: "Let us consider the wheel as

made of separate particles moving in a circle around the axis and fix

our attention on one of them. For example this particular particle. Its

mass is m1, the distance from the axis is r1. It moves in this way with

velocity v1, perpendicular to the radius vector. We write its angular

momentum. The angular momentum is L1 equal to the momentum m1, v1

transverse by r1 (L1=(m1* v1)*r1).

...

We have calculated the angular momentum of the particle. We could do

the same for this, or this, or this, and so for any particle of the

wheel. And, having done this, the total angular impulse of the wheel is

found by adding together all these different angular impulses".

But your simulation animation seems to confirm exactly why it

preccesses and which direction it must take, rather than rule it out.

To start with when the wheel rotates freely your two particles

take very different path lengths. From your animation I measured

E as being 17.5 cm And Z as being 25.5 cm. (And incidentally

E&Z both only take 1 path each . Not multiple paths!)

The reason for the precession seems simple. Let's study the 1/4 rotation

paths of each particle as E moves from 3:00 to 6:00 and Z moves

from 9:00 to 12:00

E starts off moving downwards. It has gravitational pull G added to

rotational momentum R.

So it speeds up.

Z on the other hand starts off moving upwards. It also

has gravitational pull G and rotational momentum R. But although R

is the same for both Z and E,..G on the other hand is opposite to

the direction of each. In the sense that G pulls on Z reducing its speed

whilst G pulls on E increasing its speed.

To compensate for these different velocities of E and Z ....Z travels

less distance because it has a slower velocity. And E travels a

greater distance as it has a greater velocity. To compensate

without distorting its shape the wheel preccesses.

As your animation confirms.

You too make the same mistake as the teacher in the video: consider

only the rotation of the wheel on its axis.

I have updated my animation

https://www.geogebra.org/m/sssuefav

adding a side view where there is another rotation highlighted with red

dashed line.

It is clearly seen that the wheel, descending by gravity, is forced to

incline following the red circumference line whose radius is the arm

AB.

This inclination is the cause of the precession and, in fact, if the

wheel did not incline, it would descend in perfect vertical, without

going either to the right or to the left.

Obviously, it depends on the principle of conservation of angular

momentum, as in the case of the ice skater who rotates faster when

bringing her arms towards her body and slower when moving them away.

In my animation, the upper half of the wheel (moving away from the axis

of rotation) slows its rotation to the right (like the skater spreading

her arms) and the lower half (moving towards the axis of rotation) it

accelerates its rotational motion to the left like the skater narrowing

her arms.

As a result, the wheel moves to the left.

If the rotation is counterclockwise, the reverse occurs and the

precession goes to the right.

All of this can be used to establish an alternative method to the

right-hand rule: the direction of precession always goes to the same

side as the particles at the bottom of the wheel.

In the clockwise spinning wheel, its bottom particles go to the left

and the precession also goes to the left, in the counterclockwise one,

for the same reason, the precession goes to the right.